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Semiorthogonal Decompositions and Birational Geometry of Del Pezzo Surfaces Over Arbitrary Fields

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Semiorthogonal Decompositions and Birational Geometry of Del Pezzo Surfaces Over Arbitrary Fields SEMIORTHOGONAL DECOMPOSITIONS AND BIRATIONAL GEOMETRY OF DEL PEZZO SURFACES OVER ARBITRARY FIELDS ASHER AUEL AND MARCELLO BERNARDARA Abstract. We study the birational properties of geometrically rational surfaces from a derived categorical perspective. In particular, we give a criterion for the rationality of a del Pezzo surface S over an arbitrary field, namely, that its derived category decomposes into zero- dimensional components. For deg(S) ≥ 5, we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of S from Brauer classes and Chern classes associated to these vector bundles. Introduction In their address to the 2002 International Congress of Mathematicians in Beijing, Bondal and Orlov [31] suggest that the bounded derived category Db(S) of coherent sheaves on a smooth projective variety S could provide new tools to explore the birational geometry of S, in particular via semiorthogonal decompositions. The work of many authors [72], [75], [20], [11], [1] now provides evidence for the usefulness of semiorthogonal decompositions in the birational study of complex projective varieties of dimension at most 4. A survey can be found in [77]. At the same time, the relevance of semiorthogonal decompositions to other areas of algebraic and noncommutative geometry has been growing rapidly, as Kuznetsov [76] points out in his address to the 2014 ICM in Seoul. In this context, based on the classical notion of representability for Chow groups and motives, Bolognesi and the second author [21] proposed the notion of categorical representability (see Definition 1.14), and formulated the following question. Question 1. Is a rational variety always categorically representable in codimension 2? When the base field k is not algebraically closed, the existence of k-rational points on S (being a necessary condition for rationality) is a major open question in arithmetic geometry. We are indebted to H. Esnault, who posed the following question to us in 2009. Question 2. Let S be a smooth projective variety over a field k. Can the bounded derived category Db(S) of coherent sheaves detect the existence of a k-rational point on S? This question, which was the original motivation for the current work, is now central to a growing body of research into arithmetic aspects of the theory of derived categories, see [3], [8], [11], [12], [79], [55], [109], [10]. As an example, if S is a smooth projective surface, Hassett and Tschinkel [55, Lemma 8] prove that the index of S can be recovered from Db(S) as the greatest common divisor of the second Chern classes of objects. Recall that the index ind(S) of a variety S over k is the greatest common divisor of the degrees of closed points of S. These questions also intertwine with concurrent developments in equivariant and descent aspects of triangulated and dg categories in the context of noncommutative geometry, see [7], [47], [74], [97], [105]. In the present text, we study these two questions for geometrically rational surfaces over an arbitrary field k, with special attention paid to the case of del Pezzo surfaces of Picard rank 1. The k-birational classification of such surfaces was achieved by Enriques, Manin [82], and Iskovskikh [57]; the study of arithmetic properties (e.g., existence of rational points, stable rationality, Hasse principle, Chow groups) has been an active area of research since the 1970s, see [38] for a survey. 1 2 ASHER AUEL AND MARCELLO BERNARDARA One of the most important invariants of a geometrically rational surface S over k is the s N´eron{Severi lattice NS(Sks ) = Pic(Sks ) of Sks = S ×k k as a module over the absolute Galois s s group Gk = Gal(k =k), where k is a fixed separable closure of k. The structure of this Galois lattice controls the exceptional lines on S, and hence the possible birational contractions. As the canonical bundle !S is always defined over k, one often considers the orthogonal complement ? !S . For del Pezzo surfaces, this lattice is a twist of a semisimple root lattice with the Galois action factoring through the Weyl group, see [70, Thm. 2.12]. C. Vial [109] has recently studied how one can, given a k-exceptional collection on a surface S, obtain information on the N´eron{ Severi lattice. In particular, he shows that a geometrically rational surface with an exceptional collection of maximal length is rational. In this paper, we deal with the opposite extreme, of surfaces of small Picard rank. Even when S has Picard rank 1, and there are no exceptional curves on S defined over k, our perspective is that the missing information concerning the birational geometry of S can be filled in, to some extent, by considering higher rank vector bundles on S that are generators of canonical components of Db(S). For example, let S be a quadric surface with Pic(S) generated by the hyperplane section OS(1) of degree 2. It is well known that the rationality of S is equivalent to the existence of a rational point. By a result in the theory of quadratic forms, a quadric surface having a rational point is equivalent to the corresponding even Clifford algebra C0 (a quaternion algebra over the quadratic extension defining the rulings of the quadric) being split over its center. Given a rational point x 2 S(k), the Serre construction yields a rank 2 vector bundle W as an 1 1 extension of Ix by OS(−1). On the other hand, the surface Sks = Pks × Pks carries two rank one spinor bundles, corresponding to O(1; 0) and O(0; 1) (see, e.g., [87]). The vector bundle Wks is isomorphic to O(1; 0) ⊕ O(0; 1), as the point x is the complete intersection of lines in two different rulings. Not assuming the existence of a rational point, such rank 2 vector bundles are only defined ´etalelocally; the obstruction to their existence is the Brauer class of C0. There do, however, exist naturally defined vector bundles V of rank 4 on S, via descent of W ⊕ W , and satisfying End(V ) = C0. One could say that V \controls" the birational geometry of S in a noncommutative way. Derived categories and their semiorthogonal decompositions provide a natural setting for a noncommutative counterpart of the N´eron{Severi lattice with its Galois action. One can P2 i consider the Euler form χ(A; B) = i=0 dim Ext (A; B) on the derived category, and the χ- semiorthogonal systems of simple generators, the so-called exceptional collections. Over ks , these are known to exist and have interesting properties for rational surfaces. In this paper, we consider the descent properties of such collections and show how they indeed give a complete way to interpret the link between vector bundles, semisimple algebras, rationality, and rational points. In particular, on a geometrically rational surface S, the canonical line bundle !S is ? naturally an element of such a system. Then one can consider the subcategory h!Si , whose object are χ-semiorthogonal to !S. In practice, it can be more natural to consider the category ? ? AS = hOSi , which is equivalent to h!Si . We describe decompositions (or indecomposability) of this subcategory by explicit descent methods for vector bundles. In this context, certain semisimple k-algebras naturally arise and control the birational geometry of S. These algebras also provide a presentation of the K-theory of S. A sample result of this kind, that does not seem to be contained in the literature, is as follows. Let S be a del Pezzo surface of degree 5 over a field k. It is known that S is uniquely determined by an ´etalealgebra l of degree 5 over k, see [96, Thm. 3.1.3]. We prove a k- equivalence Db(S) = Db(A), where A is a finite dimensional algebra whose semisimplification is k × k × l. In particular, there is an isomorphism ∼ Ki(S) = Ki(k) × Ki(k) × Ki(l) in algebraic K-theory, for all i ≥ 0. In the context of computing the algebraic K-theory of geometrically rational surfaces, this adds to the work of Quillen [91] for Severi{Brauer surfaces (i.e., del Pezzo surfaces of degree 9), Swan [98] and Panin [88] for quadric surfaces and involution surfaces (i.e., minimal del Pezzo surfaces of degree 8), and Blunk [25] for del Pezzo surfaces of DEL PEZZO SURFACES 3 degree 6. Our method provides a uniform way to compute the algebraic K-theory of all del Pezzo surfaces degree ≥ 5. S. Gille [49] recently studied the Chow motive with integer coefficients of a geometrically rational surface, showing that they are zero dimensional if and only if the surface has a zero cycle of degree one and the Chow group of zero-cycles is torsion free after base changing to the function field. Tabuada (cf. [102]) has shown how the Morita equivalence class of the derived category of a smooth projective variety (with its canonical dg-enhancement) gives the noncommutative motive of the variety. We refrain here from defining the additive, monoidal, and idempotent complete category of noncommutative motives, whose objects are smooth and proper dg categories up to Morita equivalence; the interested reader can refer to [102]. We recall only that any semiorthogonal decomposition gives a splitting of the noncommutative motive (see [101]) and that, as well as Chow motives, noncommutative motives can be defined with coefficients in any ring R. Working with Q-coefficients, there is a well established correspondence between noncommutative and Chow motives, see [103]. Roughly speaking, the category of Chow motives embeds fully and faithfully into the category of noncommutative motives, if one \forgets" the Tate motive.
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